Random flag complexes are a natural generalization of random graphs to higher dimensions, and since every simplicial complex is homeomorphic to a flag complex this puts a measure on a wide range of possible topologies.  In this talk, I will discuss the recent proof that according to the Erdős–Rényi measure, asymptotically almost all \(d\)-dimensional flag complexes only have nontrivial (rational) homology in middle degree \(\lfloor d/2 \rfloor\). The highlighted technique is originally due to Garland -- what he called "\(p\)-adic curvature" in a somewhat different context. This method allows one to prove cohomology-vanishing theorems by showing that certain discrete Laplacians have sufficiently large spectral gap. This reduces certain questions in probabilistic topology to questions about random matrices.

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In this talk, I will describe challenges in working with public schools and our experiences with math camps, a curriculum project, and teacher training. This discussion will outline steps in setting up collaborations, how they can benefit all the parties, and problems that can occur. I will also describe lessons learned, and where we are now in some of our current projects.

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Statistical modeling amounts to specifying a set of candidates for what the probability distribution of an observed random quantity might be. Many models used in practice are of an algebraic nature in that they are defined in terms of a polynomial parametrization. The goal of this talk is to exemplify how techniques from computational algebraic geometry may be used to solve statistical problems that concern algebraic models. The focus will be on applications in hypothesis testing and parameter identification, for which we will survey some of the known results and open problems.

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One of the major developments in number theory during the 20th century is the invention of class field theory. In this talk we will recall some highlights of this theory, and explain how to apply it to solve geometric problems. In particular, we will show how to efficiently compute isogenies between elliptic curves and how to construct elliptic curves of prescribed order. Many examples will be given.

The Schramm-Loewner evolution (SLE) is a family of random fractal curves that arise in a natural manner as scaling limits of interfaces in planar critical lattice models from statistical mechanics. The SLE curves exhibit many interesting geometric structures also present in the related models, now rigorously accessible through a combination of complex analytic and probabilistic techniques. I will give an overview of some known properties and open questions regarding the fine geometry of SLE curves. The talk is in part based on joint works with G. Lawler, with S. Rohde and C. Wong, and with T. Alberts and I. Binder.

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In an inverse boundary value problem one is interested in determining the internal properties of a medium by making measurements on the boundary of the medium. In mathematical terms, one wishes to recover the coefficients of a partial differential equation inside the medium from the knowledge of the Cauchy data of the solutions on the boundary. These problems have numerous applications, ranging from medical imaging to exploration geophysics. We shall discuss some recent progress in the analysis of inverse problems for elliptic equations, starting with the celebrated Calderon problem. The case of inverse problems with rough coefficients and with measurements performed only on a portion of the boundary will also be addressed.

Various notions of the "slope" of a real-valued function pervade optimization, and variational mathematics more broadly. In the semi-algebraic setting - an appealing model for concrete variational problems - the slope is particularly well-behaved. This talk sketches a variety of surprising applications, illustrating the unifying power of slope. Highlights include error bounds for level sets, the existence and regularity of steepest descent curves in complete metric spaces (following Ambrosio et al.), transversality and the convergence of von Neumann's alternating projection algorithm, and the geometry underlying nonlinear programming active-set algorithms. This talk will be self-contained, requiring no familiarity with variational analysis, optimization theory, or semi-algebraic geometry. Joint work with A. Daniilidis (Barcelona), A.D. Ioffe (Technion), M. Larsson (Lausanne), A.S. Lewis (Cornell).

For bin packing, the input consists of n items with sizes s_1,...,s_n in [0,1] which have to be assigned to a minimum number of bins of size 1. The seminal Karmarkar-Karp algorithm from 1982 produces a solution with at most OPT + O(log^2 OPT) bins in polynomial time, where OPT denotes the value of the optimum solution.

I will describe the first improvement in 3 decades and show that one can find a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation using the Entropy Method from discrepancy theory.

I will also survey other results of mine from discrete mathematics and combinatorial optimization concerning: an approach to the Hirsch conjecture on the diameter of polytopes; the Chvatal rank of polytopes; the extension complexity of 0/1 polytopes; and the approximability of the Steiner tree problem.

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An exotic sphere is a smooth manifold which is homeomorphic, but not diffeomorphic, to a standard sphere. In which dimensions do there exist exotic spheres? I will discuss what we know about the answer to this question in terms of the classical work of Kervaire and Milnor, the recent solution of the Kervaire invariant problem by Hill, Hopkins, and Ravenel, and a current project with Hill, Hopkins and Mahowald, where we use topological modular forms to detect exotic spheres.

The operations of time shift (ƒ(t) → ƒ(t+1)) and frequency shift (ƒ(t) → e2πiωtƒ(t)) are fundamental ingredients of applied Fourier analysis, and the group of operators on L2(ℝ) that they generate gives a unitary representation of the so-called discrete Heisenberg group. How does this representation de- compose into irreducible representations? The answer provides illustrations of (i) some useful tools of modern harmonic analysis, when ω is rational, and (ii) some pathological phenomena from the dark side of representation theory, when ω is irrational. We shall discuss these results after providing a bit of background on unitary representation theory.

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In this talk we'll look into how modelers use and compromise mathematics in industry, specifically finance, internet advertising and social media, and education. The goal of the talk is to encourage mathematicians to be more engaged with the way math is used by and against the public.

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Let \({H_i}, i=1,2,...,k\) be a finite collection of finite graphs. Let \(v_i\) be the number of vertices in \(H_i\). We want to count the number of graphs G with N vertices in which \(H_i\) appears roughly \(c_i N_i^v\) times. The tools are a combination of large deviations in probability theory and Szemeredi's regularity theorem for dense graphs.

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A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability. A hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such Kakeya sets. Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Talk based on joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler).

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